Problem: Simplify and expand the following expression: $ \dfrac{1}{3p + 27}- \dfrac{3}{p - 9}+ \dfrac{5p}{p^2 - 81} $
First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $3$ out of denominator in the first term: $ \dfrac{1}{3p + 27} = \dfrac{1}{3(p + 9)}$ We can factor the quadratic in the third term: $ \dfrac{5p}{p^2 - 81} = \dfrac{5p}{(p + 9)(p - 9)}$ Now we have: $ \dfrac{1}{3(p + 9)}- \dfrac{3}{p - 9}+ \dfrac{5p}{(p + 9)(p - 9)} $ The least common multiple of the denominators is: $ 3(p + 9)(p - 9)$ In order to get the first term over $3(p + 9)(p - 9)$ , multiply by $\dfrac{p - 9}{p - 9}$ $ \dfrac{1}{3(p + 9)} \times \dfrac{p - 9}{p - 9} = \dfrac{p - 9}{3(p + 9)(p - 9)} $ In order to get the second term over $3(p + 9)(p - 9)$ , multiply by $\dfrac{3(p + 9)}{3(p + 9)}$ $ \dfrac{3}{p - 9} \times \dfrac{3(p + 9)}{3(p + 9)} = \dfrac{9(p + 9)}{3(p + 9)(p - 9)} $ In order to get the third term over $3(p + 9)(p - 9)$ , multiply by $\dfrac{3}{3}$ $ \dfrac{5p}{(p + 9)(p - 9)} \times \dfrac{3}{3} = \dfrac{15p}{3(p + 9)(p - 9)} $ Now we have: $ \dfrac{p - 9}{3(p + 9)(p - 9)} - \dfrac{9(p + 9)}{3(p + 9)(p - 9)} + \dfrac{15p}{3(p + 9)(p - 9)} $ $ = \dfrac{ p - 9 - 9(p + 9) + 15p} {3(p + 9)(p - 9)} $ Expand: $ = \dfrac{p - 9 - 9p - 81 + 15p}{3p^2 - 243} $ $ = \dfrac{7p - 90}{3p^2 - 243}$